Two opposite pages of a book or card, together with the connecting hinge in between, form a system with one internal degree of freedom. This internal degree of freedom corresponds to the opening of the page, and will be referred to as the `opening pages`, noting that this can apply to books as well as cards or related promotional articles.
A Pop-up is characterised by the fact that apart from the opening pages there are also devices present that enable a three-dimensional or motional effect to be made. This effect can add artistic or educational value to the product. The most common device to this effect is the system of hinges.
A commonly applied principle in pop-ups is based on connecting panels of sheet material to the opening pages with the aid of hinges, in such a way that the total system, including the opening pages, has one internal degree of freedom. Thus, the action of opening the pages is coupled to the action of other panels in the system. The joint actions of the said panels compose the desired pop-up effect. In some cases, there are more than one degrees of freedom. In such a case, extra effects are actuated by suitable means other than the opening of the pages.
Many embodiments of the above principle are possible. Because of the large number of possible positionings of panels and hinges and possible combinations with other panels and hinges in the system, a large number of three dimensional or motional effects are possible. On the other hand, limitations are imposed by the available space and the lack of rigidity of the sheet material. The pop-up designer faces the challenge to create surprising effects in spite of these limitations.
Especially spectacular pop-up effects are achieved if the relative distances travelled by parts of the system are large. Indeed, in this case the difference with non-pop-ups is the most clear. The distance travelled by a point on a plane rotating about a hinge, is determined by the angle of rotation and the distance from the point to the hinge. In the case of the simplest pop-up system, a panel is connected to the opening pages, and the opening pages cover the panel when they are in the closed state. In this case, the maximum distance to a hinge that a point on the panel can have is determined by the maximum distance that can be covered with the pages. Furthermore, the angle of rotation is usually limited to 180 degrees. The existence of a maximum distance of a point on a panel to the hinge, and a maximum rotation angle leads to a maximum amount of movement relative to the opening pages by any point, when said simple pop-up system is used.
A method for achieving larger movements the the above, consists of connecting a second panel of sheet material to the first panel by means of a hinge, and whereby the first panel is in turn connected to the opening pages. The second panel should be positioned in the closed state of the pop-up in such a way that it is covered by the opening pages. During the pop-up action, the movement of of the second panel is the sum of its movement relative to the hinge connecting it to the first panel, and the movement of the first panel. This movement can be made greater than that wich uses only one panel. This principle can be extended by applying more panels, whereby the motion of the last panel the sum is of its motion around the last hinge and the motion of the preceding panels.
To achieve double- or multiple hinging pop-ups in practice however, a number of problem need to be solved. Firstly, the number of motional degrees of freedom of the system should remain one, so that the coupling to the opening pages is sustained. Secondly, the rigidity of the sheet material, often paper or cardboard, limits the possibilities. Systems of hinges that could work if the panels were ideally rigid, often turn out to be insufficiently rigid in practice. This is especially the case with the multiple hinged systems designated for making large movements. A third problem is that the hinges must operate over a sufficiently large angle, as multiple hinged systems are often folded back over large angles in order to optimally use the available space when the pop-up is in its closed state.